Statistics and the Laws of Probability

For JC math (Both H1 math and H2 math), part of the A-level examinations and syllabus is constituted by statistics. Probability is the first chapter to be introduced and is a crucial foundation for scoring. 

The law of probability states that “if the probability of an event is unknown, it can be calculated using the known probabilities of several distinct events.” 

Henceforth, to understand the rules and fundamental terms to calculate probability, this article will explain some of the key concepts to approach the A-level syllabus (9709 syllabus). 

Unions and Intersections

Union

A ∪ B is the event that either A occurs or B occurs or Both A and B occur together.

Intersect

Intersect is the event that both A and B occur. 

Probability Formula

In summary, a probability is the measure of the likelihood of an event occurring. The probability of an event A occurring, for example p (A), can be represented by the formula:

P (A) = Number of possible outcomes in event A/ Total number of possible outcomes in sample space S = n (A) / n (S) 

*Note. A sample space is all possible outcomes of an experiment, for example, tossing a coin once will will have 2 possible outcomes, head or tail. Alternatively, tossing 2 coins once could have possibilities of head head, head tail, tail head and tail tail, 4 possible outcomes.

If P (A) is = 1, it means that Event A is certain and will happen.
If P(A) is = 0, it means that Event A is impossible and will not happen.

The value of the probability would usually fall within the scale of 0 ~ 1, which indicates the likelihood of an event happening respectively.

Additional Law of Probability

With the concept of probability, it is assumed that in any random experiment, there is always uncertainty to whether an event will or will not happen. The above diagram of an union and an intersection will help to explain the additional law; 

To calculate the probability of either A or B occurring or both occurring, it requires us to calculate P(A ∪  B). 


Given that the number of outcomes in A is n(A) and the number of outcomes in B is n(B), then the union can only be calculated when:
n (A ∪  B) = n (A) + n (B) – n (A ⋂ B). 
This removes the repeating/overlapping intersect between the two probabilities. 

By inserting the above into the probability formula:
n (A ∪  B) / n (S) = n (A) / n (S)  + n (B) / n (S)  – n (A ⋂ B) / n (S) .

Which forms the additional law of probability where: 

P (A ∪  B) = P (A) + P (B) – P (A ⋂ B)

Conditional Probability

If A and B are two events, not necessarily from the same experiment, then a conditional probability is used. It assumes that A occurs, given that B has already occurred.

 

This is written as P( A | B ), and read as P ( A given B). 

P ( A given B) = n (A ⋂ B) / n (B) 

Using the probability formula,  n(A ⋂ B)/n (S)  / n(B)/ n (S)

 = P ( A ⋂ B) / P (B) 

Therefore, the conditional probability formula is defined as P( A | B ) =  P ( A ⋂ B) / P (B)

Multiplication Law of Probability

Lastly, if either of the events A and B can occur without being affected by the other, then the two events are independent. IF A and B are independent, then P(A given B has occurred) is precisely the same as P (A), since A is not affected by B. 

In this situation, instead of P (A ⋂ B) = P( A | B ) x P (B),

In an independent event, it becomes P (A ⋂ B) = P( A ) x P (B). 

Therefore, through this test of independence, the conclusion is that two events, A and B are independent if and only if P (A ⋂ B) = P( A ) x P (B).

Math Concepts

With a solid understanding of the rules of probability, it will be easier to tackle probability math questions which will test on the correct application of each of the above rules in the occurrence of an event specified by the text.

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